Textbook Question
Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.
Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 2, Problem 2.7.58
Use the definitions given in Exercise 57 to prove the following infinite limits.
lim x→1^+ 1 /1 − x=−∞
Verified step by step guidance1
Understand the problem: We need to prove that as x approaches 1 from the right (x → 1^+), the expression 1/(1-x) approaches negative infinity.
Consider the behavior of the denominator (1-x) as x approaches 1 from the right. Since x is slightly greater than 1, (1-x) becomes a small negative number.
Analyze the fraction 1/(1-x): As (1-x) becomes a small negative number, the fraction 1/(1-x) becomes a large negative number because dividing by a small negative number results in a large negative value.
Formalize the limit: For any large negative number M, we need to find a δ > 0 such that for all x satisfying 0 < x - 1 < δ, the inequality 1/(1-x) < M holds true.
Conclude the proof: By choosing δ = 1 - (1/M), we ensure that as x approaches 1 from the right, 1/(1-x) becomes less than any large negative number M, thus proving the limit is negative infinity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Limits
Infinite limits describe the behavior of a function as it approaches a certain point, where the function's value increases or decreases without bound. In this case, as x approaches 1 from the right (1+), the function 1/(1-x) tends toward negative infinity, indicating that the values of the function decrease indefinitely.
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One-Sided Limits
One-Sided Limits
One-sided limits focus on the behavior of a function as it approaches a specific point from one direction only. The notation lim x→1^+ indicates that we are considering values of x that are greater than 1, which is crucial for understanding how the function behaves as it nears the point of interest from the right side.
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Understanding the Function's Behavior
To analyze the limit lim x→1^+ 1/(1-x), it is essential to understand how the denominator behaves as x approaches 1 from the right. As x gets closer to 1, 1-x approaches 0, causing the fraction to grow larger in magnitude and negative, leading to the conclusion that the limit is negative infinity.
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Related Practice
Textbook Question
Estimate the following limits using graphs or tables.
lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4
Textbook Question
Which one of the following intervals is not symmetric about x=5?
a.(1, 9)
b.(4, 6)
c.(3, 8)
d.(4.5, 5.5)
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Textbook Question
Evaluate each limit and justify your answer.
lim x→2 (3 / 2x^5−4x^2−50)^4
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Textbook Question
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→3 −5x / √4x − 3
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Textbook Question
Determine the following limits.